| 1. mn students all have different heights. They are arranged in m > 1 rows of n > 1. In each row select the shortest student and let A be the height of the tallest such. In each column select the tallest student and let B be the height of the shortest such. Which of the following are possible: A < B, A = B, A > B? If a relation is possible, can it always be realized by a suitable arrangement of the students? | |
| 2. A is an acute angle. Show that (1 + 1/sin A)(1 + 1/cos A) > 5. | |
| 3. A triangle has no angle greater than 90o. Show that the sum of the medians is greater than four times the circumradius. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003